## Is math real?

For the discussion of math. Duh.

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cyanyoshi
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### Re: Is math real?

Cauchy wrote:It's a useful mathematical model, of course. Here on Earth, it's also sometimes useful to model the sun as going around the earth, which is why we still have the terms "sunrise" and "sunset". We model the force of gravity as 9.8 m/s^2, even though it varies from height to height and even from place to place. These are both "true enough", in that the ways that they differ from reality are usually not extreme enough to mess with whatever we're doing at the moment. The same is the case for heliocentrism: the sun also moves slightly because of the gravitational effects of the Earth and other objects, but we say "close enough" and treat it as stationary for our model of everything going around the sun. (Or maybe for space travel, you can't actually say "close enough" about something like that and their models include it, I don't know.)

In case you're curious, that assumption really is good enough for planning interplanetary missions. That's what course corrections are for, after all!

PsiCubed
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### Re: Is math real?

lorb wrote:Please don't just say my statements are wrong.

I didn't "just say" they are wrong. I've explained why I think so, and you haven't yet addressed the main point I've raised.

At least tell me why you think the example I gave is not an example of experiencing gravity. I don't understand from your answers why you think that.

Because I don't think that and I've never said that.

What I did say, is that your example misses my point. I fully admit that you can experience this narrow and specific form of gravity. But this isn't true for other forms of gravity, nor is it true for the vast majority of the physical entities in the world.

We can't see the electrons inside an atom, either.

Also do you think that it is in principle possible to falsify the current theory in gravity?

Why wouldn't it be possible? I don't really understand the question. Nor do I understand what relevance it has to the discussion at hand.

PsiCubed wrote:But I still maintain that the axiom of infinity correlates much more directly with our senses, than the notion of the earth moving around the sun. The fact that numbers never end is so freaking obvious that most children deduce it on their own.

If that is so, what sensory input would falsify the axiom of infinity? (See link above)

You can't falsify an axiom.

You can, however, falsify theories regarding the way an axiom relates to the real world.

For example, if I added an apple to a row of four apples and didn't get five apples, that would falsify the notion that the set we intuitively call "the counting numbers" is infinite.

PsiCubed wrote:Just because stuff isn't always behaving the same way doesn't mean you can't observe it. Sure it's hard to get a really good understanding on gravity just by looking at the world, but that's what science does. Collect a lot of empirical data and try deduce some laws from it.

My point is that you can't even begin this process without - first - bringing some sort of mathematical model into the mix.

Sure, you don't need mathematics to observe that a thrown stone always falls down. Yes, there are certain limited aspects of gravity which are directly observable. But the statement that "every mass attracts every other mass" isn't one of them.

By the way, if you are using "children can deduce this on their own" as a test: while they may very often be able to deduce that numbers (theoretically) go on forever after one has educated them one what numbers are and how they work, even more children figure out that stuff falls down on their own without even anyone needing to tell them. Even 3 year olds that can't count to 5 have a primitive concept of gravity.

True. But their concept of gravity is limited to the "it makes things fall down" notion. And this doesn't really change when they become older.

Every school physics teacher knows that human intuition is basically Aristotelian rather than Newtonian. It's one of the major challenges of teaching classical physics in high school.

There is also absolutely no requirement to understand something in order to experience it. I can have 0 knowledge about electricity, and still get burned badly by it. Even if you think the force that makes the earth go round the sun, and the force that makes things fall down are totally different things, they both impact your senses a lot more than any math could ever do.

Fair enough.

But without mathematics, the force that makes the earth go around the sun doesn't seem "attractive". It seems to be pulling the earth sideways.

Only once we introduce the Newtonian concept of force and do (at least some primitive verson of) some calculus, we can sort that one out.

PsiCubed
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### Re: Is math real?

Cauchy wrote:
PsiCubed wrote:1. Do you believe that the earth goes around the sun? Or do you think it is merely "a useful mathematical model"?

It's a useful mathematical model, of course. Here on Earth, it's also sometimes useful to model the sun as going around the earth, which is why we still have the terms "sunrise" and "sunset". We model the force of gravity as 9.8 m/s^2, even though it varies from height to height and even from place to place. These are both "true enough", in that the ways that they differ from reality are usually not extreme enough to mess with whatever we're doing at the moment. The same is the case for heliocentrism: the sun also moves slightly because of the gravitational effects of the Earth and other objects, but we say "close enough" and treat it as stationary for our model of everything going around the sun. (Or maybe for space travel, you can't actually say "close enough" about something like that and their models include it, I don't know.)

I agree, of-course.

Moreover, even if you include everything current science knows about how gravity works, it would still be just an approximation that is "true enough" for our purposes.

Gravity" describes some set of things in the universe. "Math" goes deeper, and attaches quantities and relations to those things (when it's used to talk about the real world). Both are obviously abstractions, but my line for "what counts as real" puts them on different sides of the divide: gravity is a real thing, and math is not.

Why?

I thoroughly enjoyed reading your well-thought observations about reality and physics, but I don't see how you get from them to this final conclusion. Since both are models and both are abstractions, why treat one of them as "real" and the other as "not real"?

The axiom of infinity does not say that "numbers never end". You get that much out of the other axioms just fine. The axiom of infinity says "there's a set that contains all the natural numbers". It's much harder to deduce that from observation, without using reasoning that leads you straight into Russell's Paradox.

Ah, I forgot that not all mathematical collections of things are sets. Thank you for reminding me of that crucial point.

At any rate, the question of whether N is a set or not, is pretty much a matter of semantics. The answer hinges on the exact definition we choose for the word "set". And since mathematicians wanted the concept of sets to include "the set of natural numbers", they've adopted a definition that accommodates this need.

This is pretty similar to the way physicists adopt words like "force" or "energy" or "gravity" to represent the abstract concepts they need. So asking "can the axiom of infinity be falsified" is similar to asking "can F=ma be falsified".

The answer to both question is no. F=ma is the very definition of the concept of "force", and the axiom-of-infinity is part of the definition of the concept of "set".

More interesting questions would be:

"Are those definitions meaningful?"

"Are they useful?"

"Are there some special or extreme cases where they become irrelevant and we need something else?"

Who knows? Maybe in 1000 years we'll find it more useful to exclude infinite sets from our vocabulary, at least for certain situations. Just like we found it useful to abandon Euclidean Geometry for certain things (like General Relativity).

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### Re: Is math real?

PsiCubed wrote:At any rate, the question of whether N is a set or not, is pretty much a matter of semantics. The answer hinges on the exact definition we choose for the word "set". And since mathematicians wanted the concept of sets to include "the set of natural numbers", they've adopted a definition that accommodates this need.

This is pretty similar to the way physicists adopt words like "force" or "energy" or "gravity" to represent the abstract concepts they need. So asking "can the axiom of infinity be falsified" is similar to asking "can F=ma be falsified".

The answer to both question is no. F=ma is the very definition of the concept of "force", and the axiom-of-infinity is part of the definition of the concept of "set".

Just a quick interjection to this interesting discussion:
F=ma is not the definition, and it is actually false in cases where the mass to which the force is applied is not constant. See wiki on Variable-Mass System. The correct definition of force is the derivative of momentum, F=d(mv)/dt.

Cauchy
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### Re: Is math real?

PsiCubed wrote:
Gravity" describes some set of things in the universe. "Math" goes deeper, and attaches quantities and relations to those things (when it's used to talk about the real world). Both are obviously abstractions, but my line for "what counts as real" puts them on different sides of the divide: gravity is a real thing, and math is not.

Why?

I thoroughly enjoyed reading your well-thought observations about reality and physics, but I don't see how you get from them to this final conclusion. Since both are models and both are abstractions, why treat one of them as "real" and the other as "not real"?

Gravity is not a model. There are models of gravity, but gravity itself is not a model, and it's only an abstraction insofar as I use the term to describe a (seemingly) disparate collection of phenomena. I can point to a thing happening in the world and say "look, gravity in action". I can't point to anything in the world and say "look, math in action". I can't point to anything in the world and say, "look, this thing is math". I can use math to inform myself about how objects and phenomena behave, but I can't single out a thing or an action and say "this is math".

That's the difference, for me, at least. If you think you can point to something and say that that thing is math, I'd sure like to know about it!
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doogly
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### Re: Is math real?

Gravity is totally a model! Or rather, Aristotelian gravity is a shit model, Newtonian gravity is a much better model, Einsteinian gravity is the most primo shit we got.

If you have names for the things you are feeling or perceiving, it is because you have models for them.
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lorb
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### Re: Is math real?

PsiCubed wrote:
lorb wrote:Also do you think that it is in principle possible to falsify the current theory in gravity?

Why wouldn't it be possible? I don't really understand the question. Nor do I understand what relevance it has to the discussion at hand.

Something that can be falsified empirically can be experienced by the senses ...

PsiCubed wrote:
lorb wrote:
PsiCubed wrote:But I still maintain that the axiom of infinity correlates much more directly with our senses, than the notion of the earth moving around the sun. The fact that numbers never end is so freaking obvious that most children deduce it on their own.

If that is so, what sensory input would falsify the axiom of infinity? (See link above)

You can't falsify an axiom.
You can, however, falsify theories regarding the way an axiom relates to the real world.
For example, if I added an apple to a row of four apples and didn't get five apples, that would falsify the notion that the set we intuitively call "the counting numbers" is infinite.

... and something that can not be falsified empirically can not be experienced by the senses.

So I guess we do agree that gravity can be felt with the senses and math can't? Mathematics just being a bunch of axioms and stuff that follows from them.
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Cauchy
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### Re: Is math real?

doogly wrote:Gravity is totally a model! Or rather, Aristotelian gravity is a shit model, Newtonian gravity is a much better model, Einsteinian gravity is the most primo shit we got.

You indeed named three models of gravity. That makes those things models, but not necessarily gravity itself.

If you have names for the things you are feeling or perceiving, it is because you have models for them.

Having a model for something is not the same thing as the thing being a model.

To put it another way, before we had the Aristotelian model of gravity, it's not like things weren't falling to the earth. So obviously something was happening before models of gravity existed. Those somethings are (among) what I call "gravity", and I contend that they are real regardless of whether we have models for them and hence real separate from their models.

PsiCubed wrote:For example, if I added an apple to a row of four apples and didn't get five apples, that would falsify the notion that the set we intuitively call "the counting numbers" is infinite.

Oh gosh, I realize I almost missed a chance to post my favorite (not quite) SMBC comic:

https://thenib.com/one-plus-one-equals- ... 3891fafd56
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gmalivuk
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### Re: Is math real?

lorb wrote:Something that can be falsified empirically can be experienced by the senses ...
This is not true. You can't experience radio waves but you can empirically falsify all sorts of statements about them, for example.
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cyanyoshi
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### Re: Is math real?

I'm pretty much of the "shut up and calculate" school of thought when it comes to these things. However after thinking about it, I would say that gravity is real in the sense that it's part of the most comprehensive verified model of the universe we've got. It's a bit like asking if dark matter or the Higgs mechanism are real. They are all great at describing the universe the way it is currently, so sure, throw them into the "real" pile.

A funny thing is that the theories assume that GR and QFT were equally good at describing things back before the Solar System was formed as they are today. Following that train of logic, whether humans can sense (or even experimentally verify) the effects of curvy space-time is irrelevant in a way. And also that would make the luminiferous aether "real" from the perspective of scientists 200 years ago even though it's not considered real today. Did reality change from then to now, and will it change in the future once supersymmetry gets sorted out?

...This is why I like the "shut up and calculate" idea.

lorb
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### Re: Is math real?

gmalivuk wrote:
lorb wrote:Something that can be falsified empirically can be experienced by the senses ...
This is not true. You can't experience radio waves but you can empirically falsify all sorts of statements about them, for example.

I probably should have clarified that in this context "experienced by the senses" is meant in it's more philosophical usage where it is synonymous with falsifiability. The reasoning is that to obtain any data to falsify anything we often have to channel whatever the subject at hand is through different media to make it noticeable to human beings. In the same way that one can only experience that a banana is yellow with light hitting it - being bounced off in a certain way, the light travelling through the air, the light later hitting your eyeballs and the eye producing an electro-chemical impulse - turning on my radio and listening to music is a way of experiencing radio waves. Doesn't really matter. Maybe this is a better phrasing: "Things that can be falsified empirically can be the source of sensory input that in some way corresponds to the thing. Things that can't be falsified can't do that."

edit: english
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rmsgrey
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### Re: Is math real?

But we can perform empirical tests that potentially falsify aspects of mathematics.

Here's a simple one: take an empty bag and a heap of pebbles. Put 2 pebbles into the bag. Put 2 more pebbles into the bag. Count the pebbles in the bag. You've just tested whether 2+2=4.

Also, according to our best theory of gravity, gravity isn't a force - it's an illusion caused by our perception of geodesics through space-time being mistaken.

lorb
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### Re: Is math real?

You haven't falsified mathematics. What you did test is how well it applies to this particular scenario. Mathematics is defined in such a way that it is non-falsifiable.

Besides that and for your particular example you can view at mathematics as a set of symbols and rules how to manipulate them. If the symbol "2" represents some amount of pebbles, and putting them into a bag represents addition, than by definition if you put "2" + "2" pebbles in the bag, whatever amount of pebbles is in the bag after that is "4" and that leaves no way that it could ever be tested/falsified because it always results in "4" independent from the empirical universe.
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rmsgrey
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### Re: Is math real?

lorb wrote:You haven't falsified mathematics. What you did test is how well it applies to this particular scenario. Mathematics is defined in such a way that it is non-falsifiable.

Besides that and for your particular example you can view at mathematics as a set of symbols and rules how to manipulate them. If the symbol "2" represents some amount of pebbles, and putting them into a bag represents addition, than by definition if you put "2" + "2" pebbles in the bag, whatever amount of pebbles is in the bag after that is "4" and that leaves no way that it could ever be tested/falsified because it always results in "4" independent from the empirical universe.

Except that my definition of "4" is not "2+2" - my definition of "4" is "((((0+1)+1)+1)+1)". My reasoning has led me to the conclusion that "2+2" should give me "4", but until I check whether the number of pebbles in the bag is "4", by comparing with my definition, I have the possibility that my reasoning has led me to an incorrect conclusion.

You are correct that what I would falsify in this instance is not mathematics, but the combination of mathematics and my mapping of mathematics to empirical reality. On the other hand, the same can be said of any experiment that supposedly potentially falsifies "gravity" - that what would be falsified is the combination of "gravity" and the belief that "gravity" applies to the situation being observed in a particular way.

Twistar
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### Re: Is math real?

But anyways, I just want to say it doesn't seem like you're all really discussing whether math is real. Rather, you're discussing what you mean by real.

The thread was created with the intent to compare mathematics to a definition of real, but instead what you are doing is coming up with definitions of real and comparing them to mathematics.

Doogly hit on this at the beginning of the thread but the sentiment got bulldozed.

lorb
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### Re: Is math real?

I think we all agree that math is real? That question has been answered "yes" and we are now discussing how exactly math is real, I think?
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cyanyoshi
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### Re: Is math real?

I have no problem calling math "real". However I fail to see why the definition of the word "real" has any mathematical significance in the first place. Regarding the original question of if math is real or is it a human invention, I say, "Why not both?"

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### Re: Is math real?

Is maths real if it can be constructed via Dedekind cut?
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### Re: Is math real?

I always get the feeling that maths is an innate fact of the universe. You don't invent formulae so much as you discover relationships between already existing values. Trigonometry, for instance, is based in the ratios of the sides of triangles, which are Real Things[citation needed].

Maths works in any base system, and always does the same things with the same operations, so that kinda points to it not being a product of any human design. The basic operations of addition and multiplication can absolutely be represented in the real world, and counting just makes sense. You could say that humanity invented maths as the study of the relationships between these values, but the actual relationships are an innate part of existence. So I guess you could say that maths is real but we had to invent a way to represent it.
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Cauchy
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### Re: Is math real?

Twistar wrote:But anyways, I just want to say it doesn't seem like you're all really discussing whether math is real. Rather, you're discussing what you mean by real.

Of course? This discussion was always going to mostly be about the definitions of "math" and "real". Once you have those, it's not too much more work to check whether the given definition of "math" has the given property "is real".

The thread was created with the intent to compare mathematics to a definition of real, but instead what you are doing is coming up with definitions of real and comparing them to mathematics.

I'd say the intent of the thread was pretty clear when the first post by the topic creator was "Is math real?" and the second post was "Math is real.".

lorb wrote:I think we all agree that math is real? That question has been answered "yes" and we are now discussing how exactly math is real, I think?

Quite a few people don't agree that math is real, me among them. At least, not under my intuition of "real", which seems to be "exists in or acts upon the physical world, in a way other than the way that all thoughts exist in the physical world since they're patterns in physical brains" but will almost certainly get refined as people bring up counterexamples.
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Twistar
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### Re: Is math real?

Now you're discussing whether math is discovered or invented which is a discussion that has been hashed many times.

I'm a pretty staunch believer that math is a pattern of thought which was invented. It is so great and powerful because it is a pattern of thought which is flexible, repeatable and easy to communicate. However, it wasn't "out there" in some sense. Humans came up with it and humans manipulate.

To put it a bit more bluntly and in terms of how some modern mathematics is done: Humans came up with (invented) the language of formal logic in which mathematics is expressed and wrote down (invented) the axioms. You could then argue that the statements which are derived from those axioms are discovered (since those relationships sort of exist as soon as you write the axioms down) and that would be fine, but it is still the case that the original pattern of thinking is a human invented construction.

Then people respond: "You don't need axioms to do math! We did math without axioms for thousands of years without them!" Fine, the argument isn't as crystal clear for this but it is still the case that math consisted of human patterns of thought. It is possible for there to be three rocks in a pond without a human having ever thought of the idea of the number three*.

To respond to Kalium_Puceon's triangle example: The triangles that we think about in math do not exist in the material world. The ideal triangles which we think about only exist in our minds. It happens to be the case that we can make a correspondence between objects in the material world and the ideal triangles in our mind, and often these correspondences and subsequent manipulations can allow us to make useful predictions and row forward the boat of human progress, but that doesn't mean perfect triangles exist in the material world.

I think the perception that math is somehow "out there" or "inate" in the universe stems from The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Math just works SO DAMN WELL to describe our physical world that people believe it must be something intrinsic to the universe. I argue that this is not the case and it is really reflective of something intrinsic to ourselves while simultaneously remarking how great it is that this thing which is ONLY intrinsic to ourselves is able to help us in our struggle to get through our lives within the material universe.

But anyways, I reiterate that this discussion has nothing to do with whether math is real or not. To determine if math is real you need to define "real". If we're done discussing whether math is real and want to talk about whether math is discovered or invented maybe we should leave this thread and make a new one more a propos to what people are trying to discuss.

*Four rocks would be a whole different matter of course
_________________________________________________________________
Responding who Cauchy who just replied before me:
Cauchy wrote:Quite a few people don't agree that math is real, me among them. At least, not under my intuition of "real", which seems to be "exists in or acts upon the physical world, in a way other than the way that all thoughts exist in the physical world since they're patterns in physical brains" but will almost certainly get refined as people bring up counterexamples.

I mean yes it's clear that under that definition math is not real. There's no a debate to be had there. Would anyone in this thread disagree with the statement that math is NOT real under that definition of real? You can easily take a definition of real that allows thoughts to be real and then math would definitely be real. The debate everyone is having is what is the "right" definition of real. It's frustrating because you can take whatever arbitrary definition of real you want to support whichever argument you want to support for whatever unrelated reason and it doesn't lead to any interesting insights about math or reality, just pissing contests over who can argue better and be more convincing, but this is the internet so of course no one is going to be convinced of anything.

Whew, that got a little bit ranty there. Sorry about that. Nothing against anyone. I guess I just think its high time we re-evaluate what should be the goal/aim of this thread/topic.

gmalivuk
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### Re: Is math real?

I would argue that math's effectiveness in the natural sciences is reasonable not because it's intrinsic to us, but because mathematics is the study of consistent systems and science works from the assumption that the universe is consistent.

Particular branches of mathematics are historically contingent and perhaps in some sense "invented", but I believe we would recognize as mathematics a wide range of possible alien endeavors, because they would also be the study of (different) consistent systems.
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Twistar
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### Re: Is math real?

gmalivuk wrote:I would argue that math's effectiveness in the natural sciences is reasonable not because it's intrinsic to us, but because mathematics is the study of consistent systems and science works from the assumption that the universe is consistent.

Particular branches of mathematics are historically contingent and perhaps in some sense "invented", but I believe we would recognize as mathematics a wide range of possible alien endeavors, because they would also be the study of (different) consistent systems.

I wasn't saying math is effective because it's intrinsic to us. I was saying that despite the fact that mathematics is so effective in the natural sciences, mathematics is not intrinsic to the natural world; rather, it is intrinsic to how we conceptualize the world.

Not sure what you're getting at with your second sentence but I think I agree. When you say "in some sense 'invented' " I think my point is that "invented" is a better choice of word than "discovered". As far as aliens: I'm not saying math is unique to humans, I'm just saying math as we understand is of our own construction and conceptualization.

Or maybe I'm missing your point. Maybe your argument is that if aliens come up with something we would call mathematics there must be something intrinsic the universe that we're all discovering? Namely the idea that the universe is consistent? Not sure. I think what I would say is that it is just different people coming up with the ideas of mathematics independently.

Just because the universe is sitting there being all consistent just waiting for someone to think of mathematics to effectively describe it doesn't mean the first person to come up with mathematical ideas discovered it rather than invented it.. That would be sort of like saying the words of Moby Dick were discovered by Herman Melville since that combination of words could have in principle been come up with once English was established.

Summary: I used to say: humans invented mathematics. Now what I say is: "invented" is a better word for describing what humans did to mathematics than "discovered".

Cauchy
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### Re: Is math real?

Twistar wrote:I mean yes it's clear that under that definition math is not real. There's no a debate to be had there. Would anyone in this thread disagree with the statement that math is NOT real under that definition of real?

I'm pretty sure that PsiCubed was arguing that under that definition, math *is* real, or at least as real as things like gravity which I would mark as real under my definition.

Twistar wrote:But anyways, I reiterate that this discussion has nothing to do with whether math is real or not.

If math is "out there" and discovered instead of invented, that's a stronger case for it being real under many definitions of real. So I'd say that that discussion has some relevance to whether math is real.

I guess I'm not sure where you'd want the discussion to go if it weren't this. I don't think this conversation was ever going to be anything other than "What do you mean by math? What do you mean by real?" once people stopped talking past each other.
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PsiCubed
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### Re: Is math real?

It's funny that the original poster of this thread had just vanished, leaving us to argue endlessly.

Me thinks we've all been nerd sniped.

To be honest, I think questions that ask "is X real" are kind of pointless. Reminds me of that famous joke about the philosopher who claimed time is an illusion, and then immediately proceeded to exclaim that he was late for meeting...

Bloopy
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Joined: Wed May 04, 2011 9:16 am UTC
Location: New Zealand

### Re: Is math real?

gmalivuk wrote:
lorb wrote:Something that can be falsified empirically can be experienced by the senses ...
This is not true. You can't experience radio waves but you can empirically falsify all sorts of statements about them, for example.

• 500mW feels almost exactly like a bee sting
• 5W feels like brushing up against a running electric stove on low power
• 50W feels like the same but on full power
• 100W feels like someone holding a lighter under your hand
• 200W feels like ohgodI'mgoingtolosemyhandhere
• 1kW, I suspect is a hospital visit with morphine and a trip to the burns unit